Assume that there is a map from a function space to another function space and a function so that is the image of i.e., A differential operator is represented as a linear combination finitely generated by and its derivatives containing higher degree such as

where the set of non-negative integers, , is called a multi-index, called length, are functions on some open domain in n-dimensional space and The derivative above is one as functions or, sometimes, distributions or hyperfunctions and or sometimes, .

In writing, following common mathematical convention, the argument of a differential operator is usually placed on the right side of the operator itself. Sometimes an alternative notation is used: The result of applying the operator to the function on the left side of the operator and on the right side of the operator, and the difference obtained when applying the differential operator to the functions on both sides, are denoted by arrows as follows:

Such a bidirectional-arrow notation is frequently used for describing the probability current of quantum mechanics.

In the functional space of square integrable functions, the scalar product is defined by

where the line over g(x) denotes the complex conjugate of g(x). If one moreover adds the condition that f or g vanishes for and , one can also define the adjoint of T by

This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When is defined according to this formula, it is called the formal adjoint of T.

A (formally) self-adjoint operator is an operator equal to its own (formal) adjoint.

Any polynomial in D with function coefficients is also a differential operator. We may also compose differential operators by the rule

Some care is then required: firstly any function coefficients in the operator D2 must be differentiable as many times as the application of D1 requires. To get a ring of such operators we must assume derivatives of all orders of the coefficients used. Secondly, this ring will not be commutative: an operator gD isn't the same in general as Dg. In fact we have for example the relation basic in quantum mechanics:

The subring of operators that are polynomials in D with constant coefficients is, by contrast, commutative. It can be characterised another way: it consists of the translation-invariant operators.

If R is a ring, let be the non-commutative polynomial ring over R in the variable D and X, and I the two-sided ideal generated by DX-XD-1, then the ring of univariate polynomial differential operators over R is the quotient ring . This is a non-commutative simple ring. Every elements can be written in a unique way as a R-linear combination of monomials of the form . It supports an analogue of the Euclidean division of polynomials.

Differential modules over (for the standard derivation) can be identified with modules over .

If R is a ring, let be the non-commutative polynomial ring over R in the variables , and I the two-sided ideal generated by the elements for all where is Kronecker delta, then the ring of multivariate polynomial differential operators over R is the quotient ring .

This is a non-commutative simple ring. Every elements can be written in a unique way as a R-linear combination of monomials of the form .

where jk: Γ(E) → Γ(Jk(E)) is the prolongation that associates to any section of E its k-jet.

This just means that for a given sections of E, the value of P(s) at a point x ∈ M is fully determined by the kth-order infinitesimal behavior of s in x. In particular this implies that P(s)(x) is determined by the germ of s in x, which is expressed by saying that differential operators are local. A foundational result is the Peetre theorem showing that the converse is also true: any (linear) local operator is differential.

An equivalent, but purely algebraic description of linear differential operators is as follows: an R-linear map P is a kth-order linear differential operator, if for any k + 1 smooth functions we have

Here the bracket is defined as the commutator

This characterization of linear differential operators shows that they are particular mappings between modules over a commutative algebra, allowing the concept to be seen as a part of commutative algebra.